Problem: What is the probability that a positive integer less than or equal to 24 is a factor of 24? Express your answer as a common fraction.
There are obviously 24 positive integers less than or equal to 24. Of them, we can count the divisors directly, or use the neat trick of prime factorizing $24 = 2^3 \cdot 3^1$. Considering that any divisor must be of the form $2^a \cdot 3^b$ where $0 \le a \le 3$ and $0 \le b \le 1$ so that there are $4 \cdot 2 = 8$ factors of 24. So, the probability of a positive integer less than or equal to 24 being a factor of 24 is $\frac{8}{24} = \boxed{\frac{1}{3}}$.